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The Mathematical Mechanic

Book Description

Everybody knows that mathematics is indispensable to physics--imagine where we'd be today if Einstein and Newton didn't have the math to back up their ideas. But how many people realize that physics can be used to produce many astonishing and strikingly elegant solutions in mathematics? Mark Levi shows how in this delightful book, treating readers to a host of entertaining problems and mind-bending puzzlers that will amuse and inspire their inner physicist.

Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. Did you know it's possible to derive the Pythagorean theorem by spinning a fish tank filled with water? Or that soap film holds the key to determining the cheapest container for a given volume? Or that the line of best fit for a data set can be found using a mechanical contraption made from a rod and springs? Levi demonstrates how to use physical intuition to solve these and other fascinating math problems. More than half the problems can be tackled by anyone with precalculus and basic geometry, while the more challenging problems require some calculus. This one-of-a-kind book explains physics and math concepts where needed, and includes an informative appendix of physical principles.

The Mathematical Mechanic will appeal to anyone interested in the little-known connections between mathematics and physics and how both endeavors relate to the world around us.

Table of Contents

  1. Cover
  2. Half title
  3. Title
  4. Copyright
  5. Contents
  6. 1 Introduction
    1. 1.1 Math versus Physics
    2. 1.2 What This Book Is About
    3. 1.3 A Physical versus a Mathematical Solution: An Example
    4. 1.4 Acknowledgments
  7. 2 The Pythagorean Theorem
    1. 2.1 Introduction
    2. 2.2 The “Fish Tank” Proof of the Pythagorean Theorem
    3. 2.3 Converting a Physical Argument into a Rigorous Proof
    4. 2.4 The Fundamental Theorem of Calculus
    5. 2.5 The Determinant by Sweeping
    6. 2.6 The Pythagorean Theorem by Rotation
    7. 2.7 Still Water Runs Deep
    8. 2.8 A Three-Dimensional Pythagorean Theorem
    9. 2.9 A Surprising Equilibrium
    10. 2.10 Pythagorean Theorem by Springs
    11. 2.11 More Geometry with Springs
    12. 2.12 A Kinetic Energy Proof: Pythagoras on Ice
    13. 2.13 Pythagoras and Einstein?
  8. 3 Minima and Maxima
    1. 3.1 The Optical Property of Ellipses
    2. 3.2 More about the Optical Property
    3. 3.3 Linear Regression (The Best Fit) via Springs
    4. 3.4 The Polygon of Least Area
    5. 3.5 The Pyramid of Least Volume
    6. 3.6 A Theorem on Centroids
    7. 3.7 An Isoperimetric Problem
    8. 3.8 The Cheapest Can
    9. 3.9 The Cheapest Pot
    10. 3.10 The Best Spot in a Drive-In Theater
    11. 3.11 The Inscribed Angle
    12. 3.12 Fermat’s Principle and Snell’s Law
    13. 3.13 Saving a Drowning Victim by Fermat’s Principle
    14. 3.14 The Least Sum of Squares to a Point
    15. 3.15 Why Does a Triangle Balance on the Point of Intersection of the Medians?
    16. 3.16 The Least Sum of Distances to Four Points in Space
    17. 3.17 Shortest Distance to the Sides of an Angle
    18. 3.18 The Shortest Segment through a Point
    19. 3.19 Maneuvering a Ladder
    20. 3.20 The Most Capacious Paper Cup
    21. 3.21 Minimal-Perimeter Triangles
    22. 3.22 An Ellipse in the Corner
    23. 3.23 Problems
  9. 4 Inequalities by Electric Shorting
    1. 4.1 Introduction
    2. 4.2 The Arithmetic Mean Is Greater than the Geometric Mean by Throwing a Switch
    3. 4.3 Arithmetic Mean ≥ Harmonic Mean for n Numbers
    4. 4.4 Does Any Short Decrease Resistance?
    5. 4.5 Problems
  10. 5 Center of Mass: Proofs and Solutions
    1. 5.1 Introduction
    2. 5.2 Center of Mass of a Semicircle by Conservation of Energy
    3. 5.3 Center of Mass of a Half-Disk (Half-Pizza)
    4. 5.4 Center of Mass of a Hanging Chain
    5. 5.5 Pappus’s Centroid Theorems
    6. 5.6 Ceva’s Theorem
    7. 5.7 Three Applications of Ceva’s Theorem
    8. 5.8 Problems
  11. 6 Geometry and Motion
    1. 6.1 Area between the Tracks of a Bike
    2. 6.2 An Equal-Volumes Theorem
    3. 6.3 How Much Gold Is in a Wedding Ring?
    4. 6.4 The Fastest Descent
    5. 6.5 Finding sin t and cos t by Rotation
    6. 6.6 Problems
  12. 7 Computing Integrals Using Mechanics
    1. 7.1 Computing by Lifting a Weight
    2. 7.2 Computing sin tdt with a Pendulum
    3. 7.3 A Fluid Proof of Green’s Theorem
  13. 8 The Euler-Lagrange Equation via Stretched Springs
    1. 8.1 Some Background on the Euler-Lagrange Equation
    2. 8.2 A Mechanical Interpretation of the Euler-Lagrange Equation
    3. 8.3 A Derivation of the Euler-Lagrange Equation
    4. 8.4 Energy Conservation by Sliding a Spring
  14. 9 Lenses, Telescopes, and Hamiltonian Mechanics
    1. 9.1 Area-Preserving Mappings of the Plane: Examples
    2. 9.2 Mechanics and Maps
    3. 9.3 A (Literally!) Hand-Waving “Proof” of Area Preservation
    4. 9.4 The Generating Function
    5. 9.5 A Table of Analogies between Mechanics and Analysis
    6. 9.6 “The Uncertainty Principle”
    7. 9.7 Area Preservation in Optics
    8. 9.8 Telescopes and Area Preservation
    9. 9.9 Problems
  15. 10 A Bicycle Wheel and the Gauss-Bonnet Theorem
    1. 10.1 Introduction
    2. 10.2 The Dual-Cones Theorem
    3. 10.3 The Gauss-Bonnet Formula Formulation and Background
    4. 10.4 The Gauss-Bonnet Formula by Mechanics
    5. 10.5 A Bicycle Wheel and the Dual Cones
    6. 10.6 The Area of a Country
  16. 11 Complex Variables Made Simple(r)
    1. 11.1 Introduction
    2. 11.2 How a Complex Number Could Have Been Invented
    3. 11.3 Functions as Ideal Fluid Flows
    4. 11.4 A Physical Meaning of the Complex Integral
    5. 11.5 The Cauchy Integral Formula via Fluid Flow
    6. 11.6 Heat Flow and Analytic Functions
    7. 11.7 Riemann Mapping by Heat Flow
    8. 11.8 Euler’s Sum via Fluid Flow
  17. Appendix. Physical Background
    1. A.1 Springs
    2. A.2 Soap Films
    3. A.3 Compressed Gas
    4. A.4 Vacuum
    5. A.5 Torque
    6. A.6 The Equilibrium of a Rigid Body
    7. A.7 Angular Momentum
    8. A.8 The Center of Mass
    9. A.9 The Moment of Inertia
    10. A.10 Current
    11. A.11 Voltage
    12. A.12 Kirchhoff’s Laws
    13. A.13 Resistance and Ohm’s Law
    14. A.14 Resistors in Parallel
    15. A.15 Resistors in Series
    16. A.16 Power Dissipated in a Resistor
    17. A.17 Capacitors and Capacitance
    18. A.18 The Inductance: Inertia of the Current
    19. A.19 An Electrical-Plumbing Analogy
    20. A.20 Problems
  18. Bibliography
  19. Index